This paper is concerned with mutational analysis found by Aubin and developed by Lorenz. To extend their results so that they can be applied to quasi-linear evolution equations initiated by Kato, we focus on a mutational framework where for each r > 0 there exists M ae 1 such that d(I(t, x), I(t, y)) ae Md(x, y) for t a [0, 1] and x, y a D (r) (phi), where I is a transition and Dr(phi) is the revel set of a proper lower semicontinuous functional phi. The setting that the constant M may be larger than 1 plays an important role in applying to quasi-linear evolution equations. In that case, it is difficult to estimate the distance between two approximate solutions to mutational equations. Our strategy is to construct a family of metrics depending on both time and state, with respect to which transitions are contractive in some sense.
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